We prove that any graph with maximum degree A sufficiently large, has a proper vertex colouring using A + 1 colours such that each colour class appears at most log 8 A times in the neighbourhood of any vertex. We also show that for ~ ~ 1, the minimum number of colours required to colour any such graph so that each vertex appears at most j3 times in the neighbourhood of any vertex is 0(A+AI+I/~/~), showing in particular that when 13=logA/loglogA, such a colouring cannot always be achieved with O(A) colours. We also provide a polynomial time algorithm to find such a colouring. This has applications to the total chromatic number of a graph.

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Abstract. The total chromatic number,

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